Intrnat. J. Math.
797
Math. Sci.
Vol. 3 No. 4 (1980) 797- 799
A LOWER BOUND ON THE NUMBER OF FINITE SIMPLE GROUPS
MICHAEL E. MAYS
Department of Mathematics
West Virginia University
Morgantown, West Virginia 26506
U.S.A.
(Received March 5, 1980)
ABSTRACT.
l{m <
Let S(n)
n: there is a (non-cyclic) simple group of order m}
Investigation of known families of simple groups provides the lower bound
S(n) >>
nl/4/log n.
KEV WORDS AND PHRASES. Spl groW, aymptotic formula.
1980 MATHEMATICAL SUBJECT CLASSIFICATION.
Primary 20 O 06.
The non-.spedialist reader should refer first to Hurley and Rudvalis (4).
Write
S(n)
l{m <
there is a simple group of order m}
n:
G is a simple group and
IGI
<
n}
DotThoff
(i),.
and S’ (n)
I{G:
Dornhoff and Spltznagel (2), and
Erdos (3) got successively better upper bounds for S(n) by refining an argument
which uses the Sylow theorems to generate a necessary criterion for a simple
group of order m to exist.
prime
plm
there is a
dlm
From the observation that S(n) <
such that d > 1 and d
1 (rood
P)}I
l{m <
n: for any
Dornhoff found that
M.E. MAYS
798
o(n) and Erds derived a complicated bound better than that of Dornhoff
S(n)
o(nl-).
but not as good as
It should be noted that in general S(n) < SW(n)
because it occasionally (in fact infinitely often) happens that non-isomorphic
simple groups of the same order exist.
We offer the following lower bound for S(n), hence for S’ (n)
S(n) >>
THEOREM.
/4/log
n.
We estimate the number of integers m < n which can be the order of a
PROOF.
simple group in one of the known families and note that in all but finitely many
cases the orders of the groups in that family are distinct.
From a llst of known families of simple groups (4, p. 708) we see that one
in the sense that for F
family dominates
i
simple group in family
i}l, F i(n)
O(F
(n)
l(n))
l{m
< n:
for any i.
m is the order of a
Fl(n)
is the number of
simple projective special linear groups of order less than n.
Thus to estimate S(n) from below, we count trlpletons (k,p,a) such that
i)
k is an integer greater than i,
2)
a is an integer
3)
p is a prime, and writing q
>_ I,
2 or p
and if p
p
a
3 and k
2 then a > i, and
we have
k
f(k,p,a)
q
k(k-l)/2
E
(ql-
l)/(k, p-l)
PSLk(q)
< n.
iffi2
Artln (5) showed that in exactly two cases distinct trlpletons give rise to
isomorphic groups, and in one case there are non-isomorphic groups of the same
k(k-l)/2 (k (k+l)/2)-1 < k 2
order in that family. Since f(k,p,a) < q
q
q
S(n) >>
m
l{m
< n:
ak 2
p
}I-
S(n) >>
r.
affil
there exists (k,p,a) satisfying i), 2), and 3) such that
Such tripletons may be counted by a triple sum, and we have
r.
E i/ak 2
kffi2p<n
i.
/ak
Constraining a and k so that n
>_ 2,
NUMBER OF FINITE SIMPLE GROUPS
log
s(n)
n/4E
n/aE log 2 12(nllak2),
(log
log 2
799
k-2
a-i
and the Prime Number Theorem using a
1 and k
2 yields S(n) >>
nl/4/log
This theorem is of interest because it has ben conjectured (3) that
o(nl-),
or even S’(n)
o(nl/3).
We have that 1/4
n.
S’ (n)
is a lower bound on the
exponent of n, and if when all simple groups are classified no new family denser
than the projective special linear groups appears, analyzing a perhaps more complicated triple sum carefully should yield the best exponent b in the estimate
S’(n)
o(nb).
ACKNOWLEDGMENT:
I wish to thank Professor Raymond Ayoub of the Pennsylvania
State University for his advice and help.
REFERENCES
i.
Dornhoff, L. Simple groups are scarce, Proc. Am. Math. Soc. 19(1968) 692-696.
2.
Dornhoff, L. and E. L. Spltznagel, Jr. Density of finite simple group orders,
Math Zeltschrlft 106(1968) 175-177.
3.
ErdDs, P. Remarks on some problems in number theory, Math. Balcanlca 4.32
(1974) 197-202.
4.
Hurley, J. and A. Rudvalls. Finite simple groups, Am. Math. Monthly 84(1977)
693-714.
5.
Artln, E. The orders of the linear groups, Comm. Pure and Appl. Math. 8(1952)
355-365.